What are these holidays anyway? Good question! In the case of Shemini Atzeret, no one really knows what the Torah intended it to be or how we should celebrate it. Jewish tradition, however, has transformed it into a festival that focuses our attention to (1) water for Israel (which includes the prayer "geshem" for rain), (2) a time to remember those we love that have died (through the Yizkor service), and (3) a day to savor the last moments of Sukkot.
Simchat Torah, we all know more about. It is a day to rejoice in the teachings and guidance of Torah with song, dancing, and most importantly, candy. To learn more about these holidays, vist My Jewish Learning: Shemini Atzeret and Simchat Torah
This website is a great resource for those times when you have the questions: how do we do this holiday, what does it mean, and how can I make the best meal possible!